How many moles of photon would contain sufficient energy to raise the temperature of 225g of water 21 to 96? Specific heat of water is` 4.18 J g^ (−1) K ^ (−1)` and frequency of light radiation used is` 2.45×10^ 9 s^ (−1)` .

Answer

To solve the problem of how many moles of photons are needed to raise the temperature of 225 g of water from 21°C to 96°C, we will follow these steps:

### Step 1: Calculate the Energy Required to Heat the Water
The energy required (Q) to raise the temperature of a substance can be calculated using the formula:
\[ Q = m \times s \times \Delta T \]
Where:
- \( m \) = mass of the water (in grams)
- \( s \) = specific heat capacity (in J/g·K)
- \( \Delta T \) = change in temperature (in °C or K)

Given:
- \( m = 225 \, \text{g} \)
- \( s = 4.18 \, \text{J/g·K} \)
- Initial temperature = 21°C
- Final temperature = 96°C
- \( \Delta T = 96 - 21 = 75 \, \text{K} \)

Now, substituting the values:
\[ Q = 225 \, \text{g} \times 4.18 \, \text{J/g·K} \times 75 \, \text{K} \]
\[ Q = 225 \times 4.18 \times 75 \]
\[ Q = 7035 \, \text{J} \]

### Step 2: Calculate the Energy per Mole of Photon
The energy (E) associated with one mole of photons can be calculated using the formula:
\[ E = N_A \times h \times \nu \]
Where:
- \( N_A \) = Avogadro's number \( (6.022 \times 10^{23} \, \text{mol}^{-1}) \)
- \( h \) = Planck's constant \( (6.626 \times 10^{-34} \, \text{J·s}) \)
- \( \nu \) = frequency of light \( (2.45 \times 10^9 \, \text{s}^{-1}) \)

Now, substituting the values:
\[ E = 6.022 \times 10^{23} \times 6.626 \times 10^{-34} \times 2.45 \times 10^9 \]
Calculating this gives:
\[ E \approx 97.727 \, \text{J/mol} \]

### Step 3: Calculate the Number of Moles of Photons Required
To find the number of moles of photons required, we can use the formula:
\[ \text{Number of moles of photons} = \frac{Q}{E} \]
Substituting the values we calculated:
\[ \text{Number of moles of photons} = \frac{7035 \, \text{J}}{97.727 \, \text{J/mol}} \]
Calculating this gives:
\[ \text{Number of moles of photons} \approx 71.9 \, \text{mol} \]

### Final Answer
Approximately 71.9 moles of photons are needed to raise the temperature of 225 g of water from 21°C to 96°C.

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